Question: What integer $n$ satisfies $0\le n<19$ and $$38574\equiv n\pmod{19}~?$$
Since 38 and 57 are both divisible by 19, so is $38000+570$.  Therefore we can say  \[38574=19(2030)+4.\]This tells us that  \[38574\equiv 4\pmod{19}\]The answer is $n=\boxed{4}$.